91Ó°ÊÓ

In calculus, the cylindrical shell method is particularly useful for solids of revolution when the axis of rotation is parallel to the axis of the function, specifically vertical rotational axes. This approach involves imagining the object being cut into thin cylindrical "shells" and then adding up the volumes of these shells. The method is best used in situations where slicing the volume perpendicularly to the axis of revolution would be complicated.

For our problem, the solid is formed by revolving the region bounded by the curve \(y = \cos x\) from \(x = 0\) to \(x = \frac{\pi}{2}\) around the \(y\)-axis. The cylindrical shell method is appropriate here because we're rotating around a vertical line.

The formula for finding the volume \(V\) of the solid using the cylindrical shells is: \[ V = \int_a^b 2\pi x (f(x) - g(x)) \, dx \] where:

• \(f(x)\) is the boundary function \(y = \cos x\)
• \(g(x)\) is the lower boundary in relation to \(f\), here \(y = 0\)
• \(a\) and \(b\) are the limits of integration, for this example \(x = 0\) to \(x = \frac{\pi}{2}\)

Setting up the integral this way helps encapsulate the interaction between the rotation and the area under the curve.

Integration by parts is a powerful technique derived from the product rule for differentiation and is used to integrate products of functions. The formula for integration by parts is given by: \[ \int u \, dv = uv - \int v \, du \] where \(u\) and \(dv\) are chosen such that differentiation and integration of them simplify the process.

In solving the volume of revolution problem using the cylindrical shell method, the integral \( \int 0^{\frac{\pi}{2}} 2\pi x \cos x \, dx \) is encountered. To integrate this using integration by parts, let \(u = x\) and \(dv = \cos x \, dx\), giving \(du = dx\) and \(v = \sin x\).

By applying the integration by parts formula, this turns into:\[ \int x \cos x \, dx = x \sin x - \int \sin x \, dx \] Calculating this from \(x = 0\) to \(x = \frac{\pi}{2}\) allows us to evaluate the definite integral necessary for the final volume.

Definite integrals are a way to find the exact area under a curve between two points and are represented by the notation \( \int_a^b f(x)\, dx \). They compute the signed area, taking the regions above the x-axis as positive and those below as negative.

In the context of volume, definite integrals can also represent the aggregated sum of incremental volumes to give the total volume of a solid. After setting up our integral through methods such as the cylindrical shell method, we use definite integrals to determine the precise volume.

For our example, the integral simplifies to compute:\[ V = 2\pi \left[ \left( \frac{\pi}{2} \cdot 1 \right) - 0 - \int_0^{\frac{\pi}{2}} \sin x \, dx \right] \] Upon simplifying each part, this evaluates to:

• The term \(x \sin x\) at the boundaries \(0\) and \(\frac{\pi}{2}\) results in \(\frac{\pi}{2}\).
• The integral of \(\sin x\) from 0 to \(\frac{\pi}{2}\) is simplified to 1.
• The calculated volume is \(\pi^2 - 2\pi\).

This gives us the exact volume of the revolved solid, reaffirming the utility of definite integrals in solving real-world geometry problems.